A fascinating limit to share with both calculus and probability students

I learned an amazing identity from Nassim Taleb earlier in the week. It has taken me a few days to understand it and I wanted to write a quick post just to get my thoughts down in paper before I forget them.

The identity is (sorry for the poor latex formatting):

\lim_{N\to\infty} e^{-N} (1 + N + N^2 /2! + \dots + N^N / N!) = 1/2

There are at least 2 things I think are interesting for students in this identity:

(1) An easy mistake would be to think that the limit is 1.

A great question to ask a student is why isn’t this expression just e^{-N} multiplied by the definition of e^{N}?

(2) The very slick probability-related proof here (scroll up to “the probabilistic way”):

The idea is to view the expression in the limit as a the probability that a Poisson process with an expected value of N has a value of N or less. Then, think of the process as being a sum of N independent Poisson processes with an expected value of 1. By the central limit theorem, this sum converges to a normal distribution with mean N, so exactly half of the distribution will be less than N (in the limit).

This probabilistic interpretation of the limit turns a really difficult computation into a neat and illuminating application of the central limit theorem.